I want to estimate parameters in a MIMIC model. I have one latent
variable (ksi), four reflexive indicators (y1, y2, y3 and y4) and four
formative indicators (x1, x2, x3, x4). Is there a way to do it in R? I
know there is the SEM library, but it seems not to be possible to
specify formative indicators, that is, observed exogenous variables
which causes the latent variable.
Marcos:
A MIMIC model seems to work fine in sem(). Here is an example of a MIMIC model which also has 4 indicators of a single latent variable and 4 covariates:
> S.sch <- var(school) > S.sch[upper.tri(var(school))] <- 0
> round(S.sch, 4)
Y1 Y2 Y3 Y4 X1 X2 X3 X4
Y1 1.3586 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Y2 1.0586 1.3815 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Y3 0.6709 0.6937 1.8192 0.0000 0.0000 0.0000 0.0000 0.0000
Y4 1.0452 1.1185 0.6584 3.6370 0.0000 0.0000 0.0000 0.0000
X1 0.4891 0.4929 0.3406 0.5244 1.1984 0.0000 0.0000 0.0000
X2 0.0011 0.0246 0.0236 0.0545 0.0177 0.2500 0.0000 0.0000
X3 -0.7325 -0.8166 -0.4524 -0.8481 -0.8759 -0.0017 4.5451 0.0000
X4 0.0614 0.0644 0.0110 0.0781 0.1070 -0.0009 -0.3961 0.1605> # n = 5198
> model.sch <- matrix(c( + 'Eta1 -> Y1', NA, 1, + 'Eta1 -> Y2', 'lambda21', NA, + 'Eta1 -> Y3', 'lambda31', NA, + 'Eta1 -> Y4', 'lambda41', NA, + 'X1 -> Eta1', 'gamma11', NA, + 'X2 -> Eta1', 'gamma12', NA, + 'X3 -> Eta1', 'gamma13', NA, + 'X4 -> Eta1', 'gamma14', NA, + 'Eta1 <-> Eta1', 'psi1', NA, + 'Y1 <-> Y1', 'theta1', NA, + 'Y2 <-> Y2', 'theta2', NA, + 'Y3 <-> Y3', 'theta3', NA, + 'Y4 <-> Y4', 'theta4', NA, + 'X1 <-> X1', 'phi11', NA, + 'X2 <-> X2', 'phi22', NA, + 'X3 <-> X3', 'phi33', NA, + 'X4 <-> X4', 'phi44', NA, + 'X1 <-> X2', 'phi12', NA, + 'X1 <-> X3', 'phi13', NA, + 'X1 <-> X4', 'phi14', NA, + 'X2 <-> X3', 'phi23', NA, + 'X2 <-> X4', 'phi24', NA, + 'X3 <-> X4', 'phi34', NA), ncol=3, byrow=TRUE)
> obs.vars.sch <- c('Y1', 'Y2', 'Y3', 'Y4', 'X1', 'X2', 'X3', 'X4')
> sem.sch <- sem(model.sch, S.sch, 5198)
> summary(sem.sch)
Model Chisquare = 77.445 Df = 14 Pr(>Chisq) = 8.4002e-11 Goodness-of-fit index = 0.99628 Adjusted goodness-of-fit index = 0.99044 RMSEA index = 0.029530 90 % CI: (0.0011724, 0.0011724) BIC = -71.451
Normalized Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max.
-3.82e+00 -4.27e-02 1.44e-05 6.41e-02 5.52e-01 2.74e+00 Parameter Estimates
Estimate Std Error z value Pr(>|z|)
lambda21 1.05267671 0.0156238 67.37643 0.00000000 Y2 <--- Eta1
lambda31 0.65931621 0.0184649 35.70637 0.00000000 Y3 <--- Eta1
lambda41 1.04965996 0.0257337 40.78937 0.00000000 Y4 <--- Eta1
gamma11 0.32691518 0.0134334 24.33606 0.00000000 Eta1 <--- X1
gamma12 0.04487985 0.0263389 1.70394 0.08839245 Eta1 <--- X2
gamma13 -0.11473656 0.0073527 -15.60472 0.00000000 Eta1 <--- X3
gamma14 -0.12574905 0.0371760 -3.38253 0.00071822 Eta1 <--- X4
psi1 0.76836697 0.0218450 35.17364 0.00000000 Eta1 <--> Eta1
theta1 0.35328543 0.0131454 26.87518 0.00000000 Y1 <--> Y1
theta2 0.26742302 0.0133470 20.03617 0.00000000 Y2 <--> Y2
theta3 1.38220283 0.0282621 48.90661 0.00000000 Y3 <--> Y3
theta4 2.52933440 0.0525526 48.12959 0.00000000 Y4 <--> Y4
phi11 1.19841396 0.0235177 50.95794 0.00000000 X1 <--> X1
phi22 0.24998279 0.0049079 50.93426 0.00000000 X2 <--> X2
phi33 4.54509222 0.0891715 50.97021 0.00000000 X3 <--> X3
phi44 0.16053813 0.0031543 50.89539 0.00000000 X4 <--> X4
phi12 0.01769075 0.0075963 2.32886 0.01986650 X2 <--> X1
phi13 -0.87588544 0.0345801 -25.32916 0.00000000 X3 <--> X1
phi14 0.10701444 0.0062625 17.08800 0.00000000 X4 <--> X1
phi23 -0.00173226 0.0147860 -0.11716 0.90673658 X3 <--> X2
phi24 -0.00087825 0.0027789 -0.31604 0.75197217 X4 <--> X2
phi34 -0.39612152 0.0130628 -30.32432 0.00000000 X4 <--> X3Iterations = 24
This example was taken from
http://statmodel.com/mplus/examples/continuous/cont2.html
and the results agree fairly closely. However, there does seem to be a problem with the RMSEA 90% confidence interval above. Thanks to John Fox for providing this package.
hope it helps,
Chuck Cleland
-- Chuck Cleland, Ph.D. NDRI, Inc. 71 West 23rd Street, 8th floor New York, NY 10010 tel: (212) 845-4495 (Tu, Th) tel: (732) 452-1424 (M, W, F) fax: (917) 438-0894
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